# Properties

 Label 2304.2.d.q Level $2304$ Weight $2$ Character orbit 2304.d Analytic conductor $18.398$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{7} +O(q^{10})$$ $$q + 4 q^{7} -2 i q^{13} -8 i q^{19} + 5 q^{25} -4 q^{31} -10 i q^{37} + 8 i q^{43} + 9 q^{49} -14 i q^{61} + 16 i q^{67} + 10 q^{73} -4 q^{79} -8 i q^{91} + 14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{7} + O(q^{10})$$ $$2q + 8q^{7} + 10q^{25} - 8q^{31} + 18q^{49} + 20q^{73} - 8q^{79} + 28q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1153.1
 1.00000i − 1.00000i
0 0 0 0 0 4.00000 0 0 0
1153.2 0 0 0 0 0 4.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
8.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.d.q 2
3.b odd 2 1 CM 2304.2.d.q 2
4.b odd 2 1 2304.2.d.a 2
8.b even 2 1 inner 2304.2.d.q 2
8.d odd 2 1 2304.2.d.a 2
12.b even 2 1 2304.2.d.a 2
16.e even 4 1 36.2.a.a 1
16.e even 4 1 576.2.a.e 1
16.f odd 4 1 144.2.a.a 1
16.f odd 4 1 576.2.a.f 1
24.f even 2 1 2304.2.d.a 2
24.h odd 2 1 inner 2304.2.d.q 2
48.i odd 4 1 36.2.a.a 1
48.i odd 4 1 576.2.a.e 1
48.k even 4 1 144.2.a.a 1
48.k even 4 1 576.2.a.f 1
80.i odd 4 1 900.2.d.b 2
80.j even 4 1 3600.2.f.m 2
80.k odd 4 1 3600.2.a.e 1
80.q even 4 1 900.2.a.g 1
80.s even 4 1 3600.2.f.m 2
80.t odd 4 1 900.2.d.b 2
112.j even 4 1 7056.2.a.bb 1
112.l odd 4 1 1764.2.a.e 1
112.w even 12 2 1764.2.k.h 2
112.x odd 12 2 1764.2.k.g 2
144.u even 12 2 1296.2.i.h 2
144.v odd 12 2 1296.2.i.h 2
144.w odd 12 2 324.2.e.c 2
144.x even 12 2 324.2.e.c 2
176.l odd 4 1 4356.2.a.g 1
208.m odd 4 1 6084.2.b.f 2
208.p even 4 1 6084.2.a.i 1
208.r odd 4 1 6084.2.b.f 2
240.t even 4 1 3600.2.a.e 1
240.z odd 4 1 3600.2.f.m 2
240.bb even 4 1 900.2.d.b 2
240.bd odd 4 1 3600.2.f.m 2
240.bf even 4 1 900.2.d.b 2
240.bm odd 4 1 900.2.a.g 1
336.v odd 4 1 7056.2.a.bb 1
336.y even 4 1 1764.2.a.e 1
336.bo even 12 2 1764.2.k.g 2
336.bt odd 12 2 1764.2.k.h 2
528.x even 4 1 4356.2.a.g 1
624.u even 4 1 6084.2.b.f 2
624.bi odd 4 1 6084.2.a.i 1
624.bm even 4 1 6084.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 16.e even 4 1
36.2.a.a 1 48.i odd 4 1
144.2.a.a 1 16.f odd 4 1
144.2.a.a 1 48.k even 4 1
324.2.e.c 2 144.w odd 12 2
324.2.e.c 2 144.x even 12 2
576.2.a.e 1 16.e even 4 1
576.2.a.e 1 48.i odd 4 1
576.2.a.f 1 16.f odd 4 1
576.2.a.f 1 48.k even 4 1
900.2.a.g 1 80.q even 4 1
900.2.a.g 1 240.bm odd 4 1
900.2.d.b 2 80.i odd 4 1
900.2.d.b 2 80.t odd 4 1
900.2.d.b 2 240.bb even 4 1
900.2.d.b 2 240.bf even 4 1
1296.2.i.h 2 144.u even 12 2
1296.2.i.h 2 144.v odd 12 2
1764.2.a.e 1 112.l odd 4 1
1764.2.a.e 1 336.y even 4 1
1764.2.k.g 2 112.x odd 12 2
1764.2.k.g 2 336.bo even 12 2
1764.2.k.h 2 112.w even 12 2
1764.2.k.h 2 336.bt odd 12 2
2304.2.d.a 2 4.b odd 2 1
2304.2.d.a 2 8.d odd 2 1
2304.2.d.a 2 12.b even 2 1
2304.2.d.a 2 24.f even 2 1
2304.2.d.q 2 1.a even 1 1 trivial
2304.2.d.q 2 3.b odd 2 1 CM
2304.2.d.q 2 8.b even 2 1 inner
2304.2.d.q 2 24.h odd 2 1 inner
3600.2.a.e 1 80.k odd 4 1
3600.2.a.e 1 240.t even 4 1
3600.2.f.m 2 80.j even 4 1
3600.2.f.m 2 80.s even 4 1
3600.2.f.m 2 240.z odd 4 1
3600.2.f.m 2 240.bd odd 4 1
4356.2.a.g 1 176.l odd 4 1
4356.2.a.g 1 528.x even 4 1
6084.2.a.i 1 208.p even 4 1
6084.2.a.i 1 624.bi odd 4 1
6084.2.b.f 2 208.m odd 4 1
6084.2.b.f 2 208.r odd 4 1
6084.2.b.f 2 624.u even 4 1
6084.2.b.f 2 624.bm even 4 1
7056.2.a.bb 1 112.j even 4 1
7056.2.a.bb 1 336.v odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}$$ $$T_{7} - 4$$ $$T_{11}$$ $$T_{17}$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$64 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$196 + T^{2}$$
$67$ $$256 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( -14 + T )^{2}$$